Forum members--
I've just posted 24 images from Villa Viiviani, which I took a couple of months ago.  You'll find them here:
The villa is a privately owned facility available for rent for weddings and other functions.   The management was very friendly, and willing to give tours but only on Monday mornings. The young man who guided me, Niccolo, didn't know that much about Twain or the time when they lived there.  However, he was genial, showed me all around, and indulged my desire to take photographs.  No one there speaks, so if you don't speak Italian, be prepared to play charades.   
The rooms that the Clemens family occupied are used for functions, and only the parlor looks to have anything like the kinds of furnishings that would have made it a residence. The solarium to the east of the first floor parlor was added after Twain's time.  It would have been a porch with perhaps a pergola or an awning over it.  
The views of Florence and the hills between Settignano and Fiesole are very picturesque. And walking through the garden, one has a sense of what it might have been like for Clemens.  A whiff of cigar smoke would go along way to emphasize that sense.  
The #10 bus will get you to Settignano, but Villa Viviani is below the village itself.  There's a bus stop a few hundred meters from the secondary driveway to the villa. The road is narrow and the traffic can be a little unnerving, so be prepared to hug one of the walls that line the road.  
The village is really charming, one of my favorites in the area.  It's a real community without any connection to the tourism that has such a big influence elsewhere around Florence.  The social club is running an interesting series of independent films, and the few cafes and restaurants cater to the locals. 
If Settignano today is anything like it was during Twain's time there, one can see why he enjoyed it.     
By the way, I didn't provide any directions or explanations about how to get to Villa di Quarto, in part because it's a bit of a hike from the bus that goes to Sesto Fiorentino. The address that's listed on line "Via di Quarto" is not accurate; it's considerably higher up on the hill on Via Pietro Dazzi.  But finding it doesn't necessarily do much for you, as Kevin and I independently discovered.  It's not open to the public.   I managed to track down Jacopo through his business address.  And while he was very gracious, I don't think he would be as happy if a lot of people began to make pilgrimages.  He runs a one-man office on site, and comes there only a few days a week from his home in Livorgno, about 100 km away.  It took us a few times to make an appointment that would work.